Australian and Canadian leaders serve the Queen of England

In Australia and Canada the government is democratically elected to serve the Queen of England.

The question arises: If they serve her, and not the people, why doesn't she just elect them herself?


Question: is the Privy council, on which the Prime Minister's serve, a public or private institution? 
Is it a secret society, in effect, that influences, or even sets, public policy without public oversight?

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